Spreading and shortest paths in systems with sparse long-range connections.

*(English)*Zbl 1062.82507Summary: Spreading according to simple rules (e.g., of fire or diseases) and shortest-path distances are studied on $d$-dimensional systems with a small density $p$ per site of long-range connections (‘small-world’ lattices). The volume $V\left(t\right)$ covered by the spreading quantity on an infinite system is exactly calculated in all dimensions as a function of time $t$. From this, the average shortest-path distance $l\left(r\right)$ can be calculated as a function of Euclidean distance $r$. It is found that $l\left(r\right)\sim r$ for

$$r<{r}_{c}={[2p{{\Gamma}}_{d}(d-1)!]}^{-1/d}log\left(2p{{\Gamma}}_{d}{L}^{d}\right)$$

and $l\left(r\right)\sim {r}_{c}$ for $r>{r}_{c}$. The characteristic length ${r}_{c}$, which governs the behavior of shortest-path lengths, diverges logarithmically with $L$ for all $p>0$.